Welcome to the forums, Widjden Bousmid.
I'd say 'My mobile stopped working'. As you suppose, 'break down' is mostly used for machines.
Actually, mobile phones rarely stop working because of technical faults. They run out of power or credit.
'My mobile's battery/credit ran out and my car broke down'.
(You can say 'cellphone' if you wish; it's mostly used by speakers of American English.)
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